Solving 2x + 3y = 35 With Y = 13: Find X Step-by-Step

Hey everyone! Ever get tangled up in a system of equations and feel like you're trying to solve a puzzle with missing pieces? Don't worry, you're not alone! Systems of equations can seem intimidating at first, but with a little bit of know-how, they become much more manageable. In this article, we're going to break down the solution to a specific system of equations, walking through each step so you can confidently tackle similar problems. We'll focus on finding the value of 'x' in the system:

• 2x + 3y = 35
• y = 13

So, grab your thinking caps, and let's dive in!

Understanding Systems of Equations

Before we jump into solving, let's make sure we're all on the same page about what a system of equations actually is. Think of it as a set of two or more equations that share the same variables. Our goal is to find the values for those variables that make all the equations in the system true simultaneously. In simpler terms, we're looking for the 'x' and 'y' values that work in both equations at the same time. This is where the magic happens, guys!

In our case, we have two equations:

1. 2x + 3y = 35
2. y = 13

The first equation involves both 'x' and 'y', while the second equation directly tells us the value of 'y'. This is a fantastic starting point because we already know one of the variables! This is a common technique you will often hear, especially when dealing with more complex mathematical scenarios. Knowing the value of 'y' simplifies our task significantly. It's like having a key piece of the puzzle already in place. Now, we can use this information to unlock the value of 'x'. This process of using the known value of one variable to find the other is a core strategy in solving systems of equations. It allows us to reduce the complexity of the problem and focus on finding one unknown at a time. Think of it as a divide-and-conquer approach to math! So, with 'y' already known, we're halfway there. The next step is to use this knowledge to crack the code and find 'x'. And that's exactly what we'll do in the next section.

The Substitution Method: Our Key Tool

Now that we understand the problem, let's introduce the method we'll use to solve it: substitution. The substitution method is a powerful technique for solving systems of equations, especially when one equation already has a variable isolated (like our 'y = 13' equation). The basic idea is to substitute the value of the isolated variable into the other equation. This eliminates one variable, leaving us with a simpler equation that we can solve for the remaining variable.

In our case, we already know that 'y = 13'. This is a gift! We can directly substitute this value into the first equation (2x + 3y = 35). This means we'll replace the 'y' in the first equation with the number '13'. This is the core of the substitution method – replacing a variable with its known value to simplify the equation. It's like swapping out a piece in a puzzle to see how the rest fits together. By substituting 'y' with '13', we transform the equation from one with two unknowns ('x' and 'y') to one with only one unknown ('x'). This makes the equation much easier to solve. It's like going from a complex maze to a straight path. The power of substitution lies in its ability to reduce complexity. By eliminating one variable, we can focus on solving for the other. This is a fundamental strategy in algebra and is used extensively in various mathematical problems. So, let's get ready to make that substitution and watch the equation simplify before our eyes! This is where the fun begins – we're about to turn a complex problem into a simple one.

Step-by-Step Solution: Finding the Value of X

Alright, let's get down to the nitty-gritty and solve for 'x'. Here's how we'll apply the substitution method, step-by-step:

Step 1: Substitute the value of y:

We know that y = 13. So, we'll substitute '13' for 'y' in the equation 2x + 3y = 35:

2x + 3(13) = 35

This substitution is the key that unlocks the solution. We've replaced 'y' with its known value, transforming the equation into a simpler form. It's like swapping a tricky symbol for a number, making the equation more concrete and manageable. This step highlights the power of substitution in simplifying complex equations. By replacing a variable with its value, we reduce the number of unknowns and pave the way for solving the equation. This is a fundamental technique in algebra and is used extensively in various mathematical contexts. The substitution allows us to focus on one variable at a time, making the problem less daunting. It's like breaking a large task into smaller, more manageable steps. Now that we've made the substitution, the next step is to simplify the equation and isolate 'x'. This is where the arithmetic skills come into play, and we'll carefully follow the order of operations to ensure we arrive at the correct solution. So, let's move on to the next step and continue our journey towards finding the value of 'x'.

Step 2: Simplify the equation:

Now, let's simplify the equation we got after substitution:

2x + 3(13) = 35

First, we'll multiply 3 by 13:

2x + 39 = 35

This multiplication is a crucial step in isolating 'x'. We're peeling away the layers of the equation, bringing us closer to the solution. It's like carefully unwrapping a present to reveal what's inside. The order of operations dictates that we perform multiplication before addition, so this step is essential for maintaining the integrity of the equation. By simplifying the equation, we're making it easier to manipulate and solve. It's like organizing a cluttered workspace to make it more efficient. Now that we've performed the multiplication, the equation is looking much cleaner. We're one step closer to isolating 'x' and finding its value. The next step will involve using inverse operations to further simplify the equation and get 'x' all by itself on one side. So, let's continue our journey and move on to the next step, where we'll isolate 'x' and finally uncover its value.

Step 3: Isolate the term with x:

To isolate the term with 'x' (which is 2x), we need to get rid of the '+ 39'. We can do this by subtracting 39 from both sides of the equation:

2x + 39 - 39 = 35 - 39

2x = -4

Subtracting 39 from both sides is a key step in isolating 'x'. It's like using a balancing scale – what we do to one side, we must do to the other to maintain the balance. This step demonstrates a fundamental principle of algebra: performing the same operation on both sides of an equation preserves its equality. By subtracting 39, we've effectively canceled out the '+ 39' on the left side, leaving us with just the term containing 'x'. This is a significant step forward in our quest to find the value of 'x'. It's like removing an obstacle from our path, making the destination clearer. Now that we've isolated the term with 'x', the next step is to get 'x' completely by itself. This will involve one more simple operation, and we'll be able to reveal the value of 'x'. So, let's move on to the final step and claim our prize – the solution to the equation.

Step 4: Solve for x:

Now we have 2x = -4. To solve for 'x', we need to divide both sides of the equation by 2:

2x / 2 = -4 / 2

x = -2

And there we have it! The value of x is -2. Dividing both sides by 2 is the final step in isolating 'x'. It's like the last piece of the puzzle falling into place, revealing the complete picture. This step utilizes the inverse operation of multiplication (division) to undo the multiplication by 2. By dividing both sides, we've effectively scaled down the equation to isolate 'x' on one side. This is a fundamental technique in algebra and is used extensively in solving equations of all types. Now that we've found the value of 'x', we've successfully solved the system of equations. It's like reaching the summit of a mountain after a challenging climb. The sense of accomplishment is well-deserved! We've taken a complex problem and broken it down into manageable steps, applying the substitution method and algebraic principles to arrive at the solution. So, with 'x = -2' in hand, we can confidently say that we've cracked the code!

Verification: Ensuring Our Solution is Correct

It's always a good idea to double-check our work to make sure our solution is correct. This is like proofreading an essay before submitting it, ensuring that there are no errors. To verify our solution, we'll substitute the values we found for 'x' and 'y' (x = -2 and y = 13) back into the original equations:

Equation 1: 2x + 3y = 35

2(-2) + 3(13) = 35

-4 + 39 = 35

35 = 35 (This is true!)

Substituting the values into the first equation and verifying that it holds true is a crucial step in ensuring the accuracy of our solution. It's like testing a key in a lock to see if it fits. This step demonstrates the importance of checking our work and not just assuming that our answer is correct. By substituting the values, we're essentially retracing our steps and confirming that the solution satisfies the original equation. This provides a high level of confidence in our answer. If the equation holds true, it means that the values we found for 'x' and 'y' are indeed the solution to the system of equations. However, we're not quite done yet. We need to verify the solution in both equations to be absolutely certain. So, let's move on to the second equation and repeat the process.

Equation 2: y = 13

This equation is straightforward, and we already know that y = 13, so it checks out!

Since our values for 'x' and 'y' satisfy both equations, we can confidently say that our solution is correct. Verification is a vital part of the problem-solving process. It's like having a safety net that catches us if we make a mistake. By taking the time to verify our solution, we can avoid errors and ensure that our answer is accurate. This step also reinforces our understanding of the problem and the solution process. It's like reviewing a map after a journey to solidify our route. So, with the verification complete, we can proudly declare that we've successfully solved the system of equations and found the correct values for 'x' and 'y'.

Conclusion: Mastering Systems of Equations

Great job, guys! You've successfully navigated through a system of equations and found the value of 'x'. We've seen how the substitution method can be a powerful tool for solving these types of problems. Remember, the key is to break down the problem into smaller, manageable steps, and don't be afraid to double-check your work.

Solving systems of equations is a fundamental skill in algebra and has applications in various fields, from science and engineering to economics and finance. It's like learning a new language – the more you practice, the more fluent you become. The ability to solve systems of equations opens doors to understanding and solving a wide range of real-world problems. It's a skill that empowers you to analyze complex situations and find solutions. So, keep practicing, keep exploring, and keep challenging yourself with new problems. The more you engage with these concepts, the more confident and proficient you'll become. Remember, math is not just about numbers and equations; it's about problem-solving and critical thinking. And you've just demonstrated your ability to do both! So, congratulations on mastering this system of equations, and keep up the great work! The world of mathematics is vast and fascinating, and there's always something new to learn and discover.

So, next time you encounter a system of equations, remember the steps we've covered here, and you'll be well on your way to finding the solution!